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arXiv:0710.1507 · Physical Review Letters

Critical Behavior of CP1 at θ=π , Haldane’s Conjecture, and the Relevant Universality Class

V. Azcoiti, Giuseppe Di Carlo, Angelo Galante

Published Jun 20, 2007• 25 citations
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Abstract

Using an approach to analyze the theta dependence of systems with a theta term we recently proposed, the critical behavior of CP(10 at theta=pi is studied. We find a region outside the strong coupling regime where Haldane's conjecture is verified. The critical line, however, does not belong to the universality class of the Wess-Zumino-Novikov-Witten model at topological coupling k=1 since it shows continuously varying critical exponents.

DOI
10.1103/physrevlett.98.257203

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Using an approach to analyze the theta dependence of systems with a theta term we recently proposed, the critical behavior of CP(10 at theta=pi is studied. We find a region outside the strong coupling regime where Haldane's conjecture is verified. The critical line, however, does not belong to the universality class of the Wess-Zumino-Novikov-Witten model at topological coupling k=1 since it shows continuously varying critical exponents.
Abstract We investigate the phase structure of a 2D lattice CP(1) model with a θ\thetaθ term. In particular, we aim to identify a critical region expected to exist along a θ=π\theta =\piθ=π line. To explore the phase structure nonperturbatively and avoid the sign problem, we employ the tensor renormalization group method. We make two improvements compared to previous tensor network studies. The first improvement involves refining the initial tensor. Specifically, we construct it using a quadrature method, which achieves higher accuracy compared to the conventional approach. The second improvement consists of analyzing the phase structure using the information on the conformal field theory, namely the central charge and the scaling dimensions, which can be accessed relatively easily via the tensor renormalization group method. Thanks to these improvements, we identify both the onset of the critical region, βc=0.5952(8)\beta _{\rm c}=0.5952(8)βc​=0.5952(8), and its universality class as the SU(2)k=1{}_{k=1}k=1​ Wess–Zumino–Witten model.
Abstract The Schwinger model, 1D quantum electrodynamics, has CP symmetry at θ = π due to the topological nature of the θ term. At zero temperature, it is known that as the fermion mass increases, the system undergoes a second-order phase transition to the CP broken phase, which belongs to the same universality class as the quantum Ising chain. In this paper, we obtain the phase diagram near the quantum critical point (QCP) in the temperature and fermion mass plane using first-principle Monte Carlo simulations, while avoiding the sign problem by using the lattice formulation of the bosonized Schwinger model. Specifically, we perform a detailed investigation of the correlation function of the electric field near the QCP and find that its asymptotic behavior can be described by the universal scaling function of the quantum Ising chain. This finding indicates the existence of three regions near the QCP, each characterized by a specific asymptotic form of the correlation length, and demonstrates that the CP symmetry is restored at any nonzero temperature, entirely analogous to the quantum Ising chain. The range of the scaling behavior is also examined and found to be particularly wide.
Phase structure of the CP(1) model in the presence of a topological θ\thetaθ-term
We numerically study the phase structure of the CP(1) model in the presence of a topological θ\thetaθ-term, a regime afflicted by the sign problem for conventional lattice Monte Carlo simulations. Using a bond-weighted Tensor Renormalization Group method, we compute the free energy for inverse couplings ranging from 0≤β≤1.10\leq \beta \leq 1.10≤β≤1.1 and find a CP-violating, first-order phase transition at θ=π\theta=\piθ=π. In contrast to previous findings, our numerical results provide no evidence for a critical coupling βc<1.1\beta_c<1.1βc​<1.1 above which a second-order phase transition emerges at θ=π\theta=\piθ=π and/or the first-order transition line bifurcates at θ≠π\theta\neq\piθ=π. If such a critical coupling exists, as suggested by Haldane's conjecture, our study indicates that is larger than βc>1.1\beta_c>1.1βc​>1.1.
Twelve years ago, Haldane formulated his famous conjecture for 1-d antiferromagnetic quantum spin chains. In the context of the 2-d O(3) model with a \\theta term, it predicts a phase transition at \\theta = \\pi, which has not yet been verified reliably. To simulate this we use the Wolff cluster algorithm together with an improved estimator for the topological charge distribution. Each cluster carries integer or half integer charge. Clusters with charge 1/2 are identified with merons. At \\theta = \\pi they are inactive, such that the mass gap vanishes. We obtain critical exponents which are consistent with predictions from the k=1 WZNW model, therefore confirming a second order phase transition.